Armitage's landmark paper Tests for linear trends in proportions [1]} is widely used in the evaluation of case-control genetic association studies [2]}, replicated toxicological bioassays [3]}, clinical dose finding studies [4]}, exposure epidemiological table data [5]}, among others. This CA-test is a scores test for weighted linear regression for the differences of proportions, organized in the simple 2-by-k table format with quantitative dose levels \(D_i\) . For the proportions \(\pi _i\)  is the common-formulated ordered alternative hypothesis \(H_1: \pi _1<\pi _2<,...,<\pi _k | \pi _1<\pi _k\)  (against \(H_0: \pi _1=\pi _2=,...,=\pi _k\) ) is somewhat understated. Rather, it is tested exactly whether the slope parameter \(b\)  of a linear regression differs from zero \(H_1:b>0\) .
For \(k\)  binomial variates \(y_i\)  with response probabilities \(p_i\)  in a randomized one-way layout with dose levels \(D_i\)  using samples size \(n_i\)  the test statistics is:
